Question

Prove that x² - 2px + 2p² is non-negative for all real values of x.

Answer 1

`x^2-2px+2p^2 \\ \\ =(x^2-2px+p^2)+p^2 \\ \\ =(x-p)^2+p^2`

Assuming
`p \in \mathbb{R}`,
`(x-p)^2 \geq 0` and
`p^2 \geq 0`, meaning that
`(x-p)^2+p^2 =x^2-2px+2p^2 \geq 0 \forall x \in \mathbb{R}`